bound – bound transitions. bound bound transitions2 einstein relation for bound- bound transitions...

Bound – Bound TransitionsBound – Bound Transitions

Bound Bound TransitionsBound Bound Transitions 22

Einstein Relation for Bound- Einstein Relation for Bound- Bound TransitionsBound Transitions Lower State i : gLower State i : gii = statistical weight = statistical weight Upper State j : gUpper State j : gjj = statistical weight = statistical weight For a bound-bound transition there are three For a bound-bound transition there are three

direct processes to consider:direct processes to consider: #1: Direct absorption: upward i #1: Direct absorption: upward i →→ j j #2: Return Possibility 1: Spontaneous transition #2: Return Possibility 1: Spontaneous transition

with emission of a photonwith emission of a photon #3: Return Possibility 2: Emission induced by #3: Return Possibility 2: Emission induced by

the radiation fieldthe radiation field


The Absorption ProcessThe Absorption Process

NNii(ν) R(ν) Rijij dω/4π = N dω/4π = Nii(ν) B(ν) Bijij I Iνν dω/4π dω/4π

RRij ij is the rate at which the transition occursis the rate at which the transition occurs

NNii(ν) is the number of absorbers cm(ν) is the number of absorbers cm-3-3 in state i in state i

which can absorb in the range (ν, ν + dν)which can absorb in the range (ν, ν + dν) The equation defines the coefficient BThe equation defines the coefficient B ijij

Transitions are not infinitely sharp so there is a Transitions are not infinitely sharp so there is a spread of frequencies φspread of frequencies φνν (the absorption profile) (the absorption profile)

Possibility 1: Upward i → j

01d


Let Us Absorb Some MoreLet Us Absorb Some More

If the total number of atoms is NIf the total number of atoms is Nii then the number of then the number of

atoms capable of absorbing at frequency ν is Natoms capable of absorbing at frequency ν is Nii(ν) = N(ν) = Niiφφνν

In going from i to j the atom absorbs photon of energy hνIn going from i to j the atom absorbs photon of energy hνijij

= E= Ejj - E - Eii

The rate that energy is removed from the beam is:The rate that energy is removed from the beam is:

aaνν is a macroscopic absorption coefficient such as is a macroscopic absorption coefficient such as ΚΚνν

4ij

i ij

ha I N B I


The Return Process: j The Return Process: j →→ i i

Spontaneous transition with the emission of Spontaneous transition with the emission of a photona photon

Probability of a spontaneous emission per Probability of a spontaneous emission per unit time unit time ≡≡ A Aijij

Emission energy rate:Emission energy rate: ρjρjν ν (spontaneous) = N(spontaneous) = Nj j AAjiji (hν (hνijij/4π) ψ/4π) ψνν

The emission profile is normalized: The emission profile is normalized:

∫ ∫ψψννdν = 1dν = 1

Going Down Once


The Return Process: j The Return Process: j →→ i i

Emission induced by the radiation fieldEmission induced by the radiation field ρjρjνν(induced) = N(induced) = Njj B Bjiji ψ ψνν I Iνν (hν (hνijij/4π)/4π) Note that the induced emission profile has Note that the induced emission profile has

the same form as the spontaneous emission the same form as the spontaneous emission profile.profile.

Spontaneous emission takes place Spontaneous emission takes place isotropically.isotropically.

Induced emission has the same angular Induced emission has the same angular distribution as Idistribution as Iνν

Going Down Twice


Assume TEAssume TE

IIνν = B = Bνν and the Boltzman Equation and the Boltzman Equation

NNii* B* Bijij B Bνν = N = Njj* A* Ajiji + N + Njj* B* Bjiji B Bνν ( ())– Upward = DownwardUpward = Downward

What did we do to get (What did we do to get ()?)?– Integrate over ν and assume that BIntegrate over ν and assume that Bνν ≠≠ f(ν) over a line width. The f(ν) over a line width. The

two pieces that depend on ν are Btwo pieces that depend on ν are Bνν (assumed a constant) and (assumed a constant) and

(φ(φνν,ψ,ψνν) but the latter two integrate to 1!) but the latter two integrate to 1!

This gives us Iν = Bν and the ratio of the state populations!

*

*

ijhj j kT

i i

N ge

N g


Now we do Some MathNow we do Some Math

Solution of () for Bν

Boltzman Equation

Substitute for Nj* and note that Ni* falls out!

*

* *

* *

*

* *

ij

ij

ij

j ji

i ij j ji

hj kT

j ii

hj kT

i jii

hj kT

i ij i jii

N AB

N B N B

gN N e

g

gN e A

gB

gN B N e B

g


Onward!Onward!

Use this to multiply the top and bottom of the previous equation

But this is just the Planck function so

1

1

ij

ij

h

i kT

j ji

ji

ji

hiji kT

j ji

ge

g B

A

BB

Bge

g B


A Revelation!A Revelation!The Intersection of Thermodynamics and QM

3

2

3

2

3

2

2 1

11

2

1

2

ij

ji

jihhkTiji kT

j ji

ji

ji

j ij

i ji

ji ji

j ij i ji

A

B hB

cBg eeg B

A h

B c

g B

g B

hA B

cg B g B


What Does this Mean?What Does this Mean?

We have used thermodynamic arguments but We have used thermodynamic arguments but these equations must be related these equations must be related onlyonly to the to the properties of the atoms and be independent of properties of the atoms and be independent of the radiation fieldthe radiation field

==> These are perfectly general equations==> These are perfectly general equations ==> These arguments predicted the existence of ==> These arguments predicted the existence of

stimulated emission.stimulated emission. NB: Stimulated (induced) emission is not NB: Stimulated (induced) emission is not

intuitively obvious. However, it is now an intuitively obvious. However, it is now an observed fact.observed fact.


The Equation of TransferThe Equation of TransferNote the Following:

4

4

4

4 4 4

( )4

iji ij

ijj ji

ijj ji

ij ij ijj ji j ji i ij

ijj ji i ij j ji

dIj I

dz

ha I N B I

hj N A

hj N B I

h h hdIN A N B I N B I

dzhdI

N A N B N B Idz


The Coefficients Are:The Coefficients Are:

Departure Coefficient

Gather the terms on I

Boltzmann Equation

giBij = gjBji

Line Absorption Coefficient usually denoted lν

*

*

*

( , , )

( , )

14

14

14

ij

ij

ij

i ei

i e

ij j jii ij

i ij

hj j j j j kT

i i i i

hij j ji j kT

i iji ij i

hij j kT

i iji

N T N Jb

N T N

h N Bl N B

N B

N b N b ge

Ni b N b g

h g B bl N B e

g B b

h bl N B e

b


The Line Source FunctionThe Line Source FunctionSource Function = j/

Divide by NjBjiψν

Use relations between A & B’s

The equation of transfer for a line!

3

2

1

2 1

1ij

j ji

i ij j ji

ji

ji

i ij

j ji

h

i kT

j

N AS

N B N B

A

BN B

N B

h

c be

b

dII S

dz


Let Us Simplify SomeLet Us Simplify Some

Assume φAssume φνν = ψ = ψνν (Not unreasonable) (Not unreasonable)

If we assume LTE then bIf we assume LTE then bii = b = bjj = 1 = 1

Then Then lν = N = NiiBBijijφφνν(1-e(1-e-hν/kT-hν/kT))

– The term (1-eThe term (1-e-hν/kT-hν/kT) is the correction for ) is the correction for stimulated emissionstimulated emission

SSνν = B = Bνν


An Alternate DefinitionAn Alternate Definition

Define BDefine Bi,ji,j as as uuννBBijij = B = Bijij 8π/c 8π/c33 (hν (hν33/(e/(ehν/kThν/kT-1))-1))

– Bi,j is the probability that an atom in lower level i Bi,j is the probability that an atom in lower level i will be excited to upper level j by absorption of a will be excited to upper level j by absorption of a photon of frequency ν = (Ephoton of frequency ν = (Ejj - E - Eii) / h) / h

– This B is defined with respect to the energy density U This B is defined with respect to the energy density U of the radiation field, not Iof the radiation field, not Iνν (=B (=Bνν) as previously.) as previously.

For this definitionFor this definition– AAjiji = (8πhν = (8πhν33/c/c33) B) Bjiji

– ggiiBBijij = g = gjjBBjiji

With Respect to the Energy Density


Now Let Us Try To Determine Now Let Us Try To Determine AAjiji, B, Bijij, and B, and Bjiji

Classical Oscillator and EM FieldClassical Oscillator and EM Field– Dimensionally correct but can be off by orders of Dimensionally correct but can be off by orders of

magnitudemagnitude QM Atom and Classical EM FieldQM Atom and Classical EM Field

– Correct Expression for BCorrect Expression for Bijij

QM Atom and a Quantized EM FieldQM Atom and a Quantized EM Field– Gives BGives Bijij, B, Bjiji, and A, and Ajiji

However, note that BHowever, note that Bijij, B, Bjiji, and A, and Ajiji are are interrelated and if you know one you know them interrelated and if you know one you know them all!all!


A Classical ApproachA Classical Approach

The probability of an event generally depends on The probability of an event generally depends on the number of ways that it can happen. Suppose the number of ways that it can happen. Suppose photons in the range (ν, ν + dν) are involved. photons in the range (ν, ν + dν) are involved. The statistical weight of a free particle: Position The statistical weight of a free particle: Position (x, x+dx) and momentum (p, p+dp) is dN/h(x, x+dx) and momentum (p, p+dp) is dN/h33 = = dxdydzdpdxdydzdpxxdpdpyydpdpzz/h/h33

Then the statistical weight per unit volume of a Then the statistical weight per unit volume of a particle with total angular momentum p, in particle with total angular momentum p, in direction dΩ, is pdirection dΩ, is p22dpdΩ/hdpdΩ/h33..


Free ParticlesFree Particles

Momentum of a photon is p = hν/c so the Momentum of a photon is p = hν/c so the statistical weight per unit volume of statistical weight per unit volume of photons in (ν, ν+dν) is νphotons in (ν, ν+dν) is ν22dνdΩ/hcdνdΩ/hc22..

This means high frequency transitions This means high frequency transitions have higher transition probabilities than have higher transition probabilities than low frequency transitions.low frequency transitions.


Semiclassical Treatment of an Semiclassical Treatment of an Excited AtomExcited Atom

Oscillating Dipole Oscillating Dipole (electric) in which the (electric) in which the electron oscillates electron oscillates about the nucleus.about the nucleus.

For the case of no For the case of no energy loss the energy loss the equation of motion is:equation of motion is:– νν00 is the frequency of is the frequency of

the oscillationthe oscillation The electric dipole The electric dipole

radiates (classically) radiates (classically) and energy is lost due and energy is lost due to the radiationto the radiation

22 2

024

d rm r

dt

22 2

3 2

2

3

e d rI

c dt


A Damped OscillatorA Damped Oscillator The energy loss leads to a further term in the The energy loss leads to a further term in the

equation of motion which slows the electronequation of motion which slows the electron

The damping force isThe damping force is

If F is small then the motion can be considered to be If F is small then the motion can be considered to be near a simple harmonic r = rnear a simple harmonic r = r00 cos (2πν cos (2πν00t)t)

2 3

3 3

2

3

e d rF

c dt

2 20

22 2 2 20 0 3

4

24 4

3

r r r

emr r r r

c


The Solution isThe Solution is

γ is called the classical damping constantγ is called the classical damping constant

Damped Harmonic Oscillator

20 0

2 22 22 20 02

cos(2 )

82.47(10 )

3

t

e

r r e t

e

m c


Electric FieldElectric Field

The field set up by the electron is proportional The field set up by the electron is proportional to the electron displacementto the electron displacement

At time t = (γ/2)At time t = (γ/2)-1-1 we get E(t) = E we get E(t) = E00/e which /e which

characterizes the time scale of the decaycharacterizes the time scale of the decay

20 0( ) cos(2 )

t

E t E e t


A Result!A Result!

The decay time must be proportional to AThe decay time must be proportional to A jiji -the -the probability of a spontaneous transition probability of a spontaneous transition downwardsdownwards

AAclassicalclassical = γ = γ– For Hα, = 6563Å; νFor Hα, = 6563Å; ν00 = c/ = c/λλ so γ so γ 5(10 5(1077) s) s-1-1

The field decays exponentially so the frequency The field decays exponentially so the frequency is not monochromatic. To get the frequency is not monochromatic. To get the frequency dependence do a Fourier analysisdependence do a Fourier analysis

The square of the spectrum is the intensity of the The square of the spectrum is the intensity of the field!field!


The Broadening ProfileThe Broadening ProfileLorentz Profile

0 22 2

0

2

0 22

0

20

0 22

0

22

0 2

22

0 2

( )

4 ( )2

4

( )4

42

( )4

2( )

4 4

2( )

4 4

I I

I

II I

2

22

0

0

2

( )4

( )4

2FWHM


More On BroadeningMore On Broadening

In QM the natural broadening arises due to In QM the natural broadening arises due to the uncertainty principlethe uncertainty principle

The uncertainty in the time in which an The uncertainty in the time in which an atom is in a state is its lifetime in that atom is in a state is its lifetime in that state.state.

Average lifetime AAverage lifetime A-1-1

Thus the uncertainty Thus the uncertainty E = hE = ht t ≈≈ Ah Ah This means that This means that ν ν A A γ γ


Oscillator StrengthsOscillator Strengths

Since γ is of the same order as A we usually Since γ is of the same order as A we usually express exact values of A in terms of the express exact values of A in terms of the classical value of γ.classical value of γ.

Oscillator Strength nOscillator Strength nupperupper →→ m mlowerlower

AAnmnm ≡ ≡ 3 (g3 (gmm/g/gnn) f) fnmnm γ γ = (g= (gmm/g/gnn) (8π) (8π22ee22νν22/m/meecc33) f) fnmnm

= (g= (gmm/g/gnn) 7.42(10) 7.42(10-22-22) ν) ν2 2 ffnmnm

BBmnmn = (πe= (πe22/m/meehν) fhν) fnmnm


Oscillator StrengthsOscillator Strengths

This is known as Kramer’s Formula. GI is the Gaunt Factor which is order 0.1 - 10

LineLine U-LU-L ff

HαHα 3-23-2 0.6410.641

HβHβ 4-24-2 0.1190.119

HγHγ 5-25-2 0.0440.044

HδHδ 6-26-2 0.0210.021

HεHε 7-27-2 0.0120.012

Balmer Series of H

36

2 2 3 3

2

3

2 2 3 5

2 1 1 1

3 3

2

32 1 1

3 3

Inm

m

m

Inm

Gf

g m n n m

g m

Gf

m n n m


The Absorption CoefficientThe Absorption Coefficient

We are now ready to determine the absorption We are now ready to determine the absorption coefficient acoefficient aνν per atom of an absorption line per atom of an absorption line centered at νcentered at ν00..

It is defined so that the probability of absorption It is defined so that the probability of absorption per unit path length of a photon is Naper unit path length of a photon is Naνν where N is where N is the number density of atoms capable of the number density of atoms capable of absorption.absorption.

Consider a beam of intensity IConsider a beam of intensity Iνν traveling across a traveling across a unit cross-sectional area. In time dt the photons unit cross-sectional area. In time dt the photons have traveled cdt.have traveled cdt.


Energy ConsiderationsEnergy Considerations

The energy removed from the beam in (ν, ν + dν) The energy removed from the beam in (ν, ν + dν) is Iis Iνν na naνν cdt. Alternately, the number of transitions cdt. Alternately, the number of transitions is Iis Iνν na naνν cdt / hν. The volume occupied by the cdt / hν. The volume occupied by the photons is cdt (unit cross sectional area).photons is cdt (unit cross sectional area).

The number of transitions per unit volume per The number of transitions per unit volume per unit time per unit frequency due to the beam is Iunit time per unit frequency due to the beam is Iνν nanaνν / hν. / hν.

Integrating over solid angle and assuming thermal Integrating over solid angle and assuming thermal equilibrium (4πIequilibrium (4πIν ν = cU= cUνν) the number of ) the number of absorptions is cUabsorptions is cUνν na naνν / hν. / hν.


OnwardOnward

We now integrate over frequency over the line We now integrate over frequency over the line profile only to obtain the number of transitions profile only to obtain the number of transitions from lower state to upper state per unit volume from lower state to upper state per unit volume per unit timeper unit time

But this is equal to UBut this is equal to Uνν B Bnmnm N so N so

0

0nm

UTransition Rate cN a d

h

UU B N cN a d

h


Line AbsorptionLine Absorption

UUνν does not does not vary across the vary across the lineline

aaνν is small is small except near νexcept near ν00: : νν00 = (E = (Enn-E-Emm) / ) / hh

UUνν/hν varies /hν varies slowly across slowly across the linethe line

The nature of line absorption

For ν = ν0

cm2 Hz

0

0

0 00

00

2

00

2

0

nm

nme

nme

UUcN a d cN a d

h h

UU B N cN a d

h

e cf a d

m h h

ea d f

m c


We are about doneWe are about done

The integral of aThe integral of aνν over ν can be thought of as the over ν can be thought of as the

total absorption cross section per atom initially total absorption cross section per atom initially in the lower state. We rewrite ain the lower state. We rewrite aνν as as

φφνν is the absorption profile and we expect it to is the absorption profile and we expect it to

follow the Lorentz profilefollow the Lorentz profile

2

0 nme

ea d f

m c

2

nme

ea f

m c


Combine the ProfileCombine the Profile

In principal fIn principal fnmnm ~ γ but there are other broadening ~ γ but there are other broadening

mechanisms.mechanisms.– Natural width (γ) is small compared to other mechanisms Natural width (γ) is small compared to other mechanisms

(Stark, Doppler, van der Waals, etc)(Stark, Doppler, van der Waals, etc)

For ν = νFor ν = ν00: a: a00 = (πe = (πe22/m/meec) (fc) (fnmnm/γ)/γ)

So

2

0 22

0

2 2

22

0

4

( )4

4

( )4

nme

a a

ea f

m c


Stimulated EmissionStimulated Emission

We have yet to account for stimulated emissionWe have yet to account for stimulated emission We usually assume emission is isotropic but if it goes as We usually assume emission is isotropic but if it goes as

BBmnmnIIνν then it correlates with I then it correlates with Iνν

We treat this as a negative absorption and reduce the We treat this as a negative absorption and reduce the value of avalue of aνν by the appropriate factor. by the appropriate factor.

In our discussion of the Einstein coefficients we found In our discussion of the Einstein coefficients we found the factor to be (1 - ethe factor to be (1 - e-(hν/kT)-(hν/kT)) so the corrected a) so the corrected aνν is is

2

(1 )h

kTnm

e

ea f e

m c

bound – bound transitions. bound bound transitions2 einstein relation for bound- bound transitions...

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